Full Parameter Estimates

Full fixed effect parameter estimates are included for all models below. Note that these are presented on the log scale. In the paper only the beta terms are included along with Bayes factors evaluating evidence in support of the point-null hypothesis relative to the alternative hypothesis (i.e. of a non-zero effect) using the Savage-Dickey density ratio, but not with intercepts or thresholds.

Cumulative models assume that responses come from categoristion of a latent continuous variable. Thus, for the cumulative models there is not a single intercept but instead the model output reports several “intercepts” which are thresholds that partition the latent continuous variable into response categories. Holding all other parameters constant, these give the cumulative log-odds of responses being within a given threshold. (To transform these into probabilities, the inverse logit should be applied to any given threshold.) As with intercepts in typical regression modelling, these are often not the focus of the research question. Thus, intercepts and thresholds are omitted from the main text but are included in full here for completeness.

Crucially, the thresholds in the cumulative models reported here are dictated by the unique scores on the outcome. In the depression, anxiety, and stress models each “intercept” typically represents scores from 0 to 42 in increments of 2 as the scale requires individual scores to be summed and multiplied by 2. However, in the short loneliness scale used in study 1, the thresholds represent scores from 3 to 9 in increments of 1. For the longer loneliness scale used in studies 2 and 3, the thresholds represent scores from 0 to 11 in increments of 1. In cases where the thresholds do not match the number of unique increments on the scale this is due to some unique values not occurring in the data set.

As the tables are explained fully in text, no further interpretation is given for each table.

Study 1

Main Model

term Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Depression
Intercept[1] -4.446 0.364 -5.173 -3.743 1 2708 4429
Intercept[2] -3.193 0.305 -3.811 -2.607 1 2786 4933
Intercept[3] -2.503 0.280 -3.081 -1.976 1 2785 4774
Intercept[4] -2.010 0.268 -2.552 -1.493 1 2728 4963
Intercept[5] -1.284 0.253 -1.788 -0.799 1 2682 4818
Intercept[6] -0.583 0.244 -1.061 -0.109 1 2829 5070
Intercept[7] -0.117 0.243 -0.584 0.362 1 2968 5409
Intercept[8] 0.395 0.243 -0.071 0.880 1 2960 5350
Intercept[9] 0.915 0.246 0.435 1.402 1 3048 5314
Intercept[10] 1.372 0.254 0.882 1.878 1 3154 4487
Intercept[11] 1.744 0.259 1.254 2.262 1 3179 5043
Intercept[12] 2.193 0.272 1.679 2.747 1 3253 5292
Intercept[13] 2.519 0.280 1.996 3.093 1 3240 5276
Intercept[14] 2.756 0.288 2.206 3.339 1 3265 5019
Intercept[15] 3.202 0.305 2.619 3.829 1 3412 5071
Intercept[16] 3.926 0.332 3.293 4.600 1 3625 5399
Intercept[17] 4.304 0.350 3.642 5.009 1 3856 5793
Intercept[18] 4.824 0.375 4.095 5.572 1 3939 5591
Intercept[19] 5.376 0.414 4.583 6.207 1 4131 5784
Intercept[20] 5.972 0.458 5.096 6.898 1 4401 6349
Intercept[21] 6.867 0.546 5.838 7.998 1 5002 6385
Lockdown Period 0.451 0.099 0.257 0.645 1 7956 6747
Total Hours 0.360 0.173 0.021 0.700 1 4170 5539
Lockdown Period by Hours -0.072 0.106 -0.275 0.135 1 10559 7063
Anxiety
Intercept[1] -3.560 0.353 -4.273 -2.905 1 2116 3968
Intercept[2] -1.703 0.284 -2.277 -1.171 1 2233 4186
Intercept[3] -0.687 0.263 -1.204 -0.181 1 2483 4792
Intercept[4] -0.051 0.260 -0.561 0.463 1 2453 4517
Intercept[5] 0.871 0.264 0.351 1.389 1 2604 4148
Intercept[6] 1.490 0.274 0.973 2.038 1 2676 4885
Intercept[7] 2.152 0.289 1.604 2.731 1 2755 4799
Intercept[8] 2.989 0.318 2.390 3.644 1 3003 4837
Intercept[9] 3.752 0.350 3.096 4.461 1 3230 5122
Intercept[10] 4.500 0.387 3.761 5.276 1 3600 5493
Intercept[11] 5.393 0.436 4.564 6.285 1 4008 5500
Intercept[12] 5.901 0.465 5.042 6.851 1 4271 5758
Intercept[13] 6.290 0.494 5.363 7.305 1 4476 5427
Intercept[14] 6.906 0.554 5.873 8.036 1 4912 5937
Intercept[15] 7.647 0.635 6.481 8.924 1 5464 6358
Intercept[16] 8.560 0.780 7.153 10.206 1 6382 5739
Intercept[17] 9.278 0.942 7.628 11.318 1 6879 5538
Intercept[18] 10.622 1.355 8.362 13.693 1 9188 6048
Lockdown Period 0.191 0.099 -0.002 0.387 1 8096 6130
Total Hours 0.333 0.174 0.001 0.683 1 3937 5259
Lockdown Period by Hours -0.081 0.107 -0.295 0.131 1 10132 6493
Stress
Intercept[1] -4.847 0.389 -5.655 -4.111 1 2445 3864
Intercept[2] -3.920 0.336 -4.621 -3.284 1 2459 4118
Intercept[3] -3.132 0.295 -3.739 -2.574 1 2422 3743
Intercept[4] -2.383 0.265 -2.917 -1.872 1 2599 4366
Intercept[5] -1.686 0.245 -2.180 -1.214 1 2670 4433
Intercept[6] -0.982 0.233 -1.450 -0.528 1 2786 4657
Intercept[7] -0.228 0.227 -0.679 0.222 1 2885 4379
Intercept[8] 0.433 0.227 -0.004 0.877 1 2989 4804
Intercept[9] 0.955 0.231 0.513 1.423 1 3110 4610
Intercept[10] 1.551 0.238 1.089 2.042 1 3119 4661
Intercept[11] 2.149 0.253 1.674 2.672 1 3249 4851
Intercept[12] 2.577 0.264 2.082 3.125 1 3281 5248
Intercept[13] 3.174 0.285 2.640 3.762 1 3468 5238
Intercept[14] 3.697 0.305 3.125 4.333 1 3529 5567
Intercept[15] 4.247 0.333 3.622 4.913 1 3656 4813
Intercept[16] 4.699 0.357 4.028 5.419 1 3719 5324
Intercept[17] 4.915 0.369 4.213 5.670 1 3694 5461
Intercept[18] 5.947 0.449 5.094 6.836 1 4257 5760
Intercept[19] 6.840 0.555 5.823 7.983 1 5261 5385
Intercept[20] 7.734 0.709 6.458 9.272 1 6499 6294
Intercept[21] 9.302 1.179 7.397 11.992 1 9224 6164
Lockdown Period 0.388 0.097 0.197 0.576 1 7318 6308
Total Hours 0.116 0.157 -0.195 0.423 1 3521 5316
Lockdown Period by Hours 0.053 0.104 -0.151 0.258 1 8079 5886
Loneliness
Intercept[1] -2.531 0.288 -3.115 -1.998 1 2953 4888
Intercept[2] -1.133 0.247 -1.631 -0.666 1 2962 4979
Intercept[3] 0.126 0.240 -0.345 0.597 1 3129 5454
Intercept[4] 1.886 0.270 1.372 2.428 1 3303 4045
Intercept[5] 3.218 0.316 2.628 3.867 1 3614 5057
Intercept[6] 4.443 0.384 3.721 5.230 1 3932 4907
Lockdown Period 0.620 0.108 0.410 0.833 1 8316 5558
Total Hours 0.013 0.171 -0.328 0.341 1 5076 6245
Lockdown Period by Hours 0.073 0.113 -0.146 0.299 1 10314 6402

Moderation Model

term Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Depression
Intercept[1] 1.169 0.448 0.277 2.056 1 10173 6821
Intercept[2] 1.862 0.437 1.016 2.743 1 9830 7187
Intercept[3] 2.396 0.436 1.549 3.278 1 9306 7066
Intercept[4] 2.732 0.441 1.872 3.619 1 9352 7190
Intercept[5] 3.037 0.448 2.166 3.941 1 9303 7335
Intercept[6] 3.504 0.462 2.615 4.425 1 9249 7092
Intercept[7] 3.799 0.470 2.891 4.737 1 9190 6928
Intercept[8] 4.155 0.480 3.233 5.115 1 9277 6991
Intercept[9] 4.440 0.490 3.495 5.425 1 9380 6936
Intercept[10] 4.686 0.499 3.726 5.685 1 9347 6865
Intercept[11] 4.913 0.508 3.938 5.928 1 9325 6815
Intercept[12] 5.194 0.518 4.212 6.230 1 9394 6826
Intercept[13] 5.428 0.524 4.442 6.485 1 9528 7160
Intercept[14] 5.534 0.526 4.535 6.604 1 9523 6992
Intercept[15] 5.911 0.540 4.895 6.997 1 9460 6870
Intercept[16] 6.358 0.555 5.299 7.472 1 9644 6438
Intercept[17] 6.647 0.566 5.557 7.775 1 9858 6458
Intercept[18] 6.929 0.576 5.821 8.099 1 9972 6734
Intercept[19] 7.387 0.595 6.241 8.593 1 10210 6061
Intercept[20] 7.599 0.605 6.455 8.810 1 10389 6709
Intercept[21] 8.132 0.642 6.902 9.408 1 10616 6687
Difference in Hours Played 0.013 0.022 -0.028 0.057 1 10203 5551
Loneliness During Lockdown 0.699 0.079 0.550 0.855 1 9632 6605
Hours by Loneliness -0.001 0.004 -0.008 0.006 1 10067 5659
Anxiety
Intercept[1] 0.660 0.432 -0.177 1.520 1 9896 7050
Intercept[2] 1.498 0.437 0.657 2.365 1 9594 6663
Intercept[3] 2.100 0.445 1.257 2.980 1 9397 6743
Intercept[4] 2.369 0.453 1.513 3.281 1 9295 7151
Intercept[5] 2.759 0.465 1.883 3.696 1 9092 7253
Intercept[6] 2.944 0.470 2.056 3.891 1 9030 7200
Intercept[7] 3.286 0.481 2.361 4.257 1 9100 7151
Intercept[8] 3.761 0.494 2.809 4.760 1 9122 7015
Intercept[9] 4.197 0.507 3.216 5.214 1 8848 6441
Intercept[10] 4.556 0.520 3.548 5.609 1 8996 6610
Intercept[11] 5.139 0.550 4.077 6.259 1 8796 6870
Intercept[12] 5.567 0.572 4.478 6.728 1 9499 6926
Intercept[13] 5.841 0.593 4.714 7.038 1 10120 6930
Intercept[14] 6.473 0.650 5.246 7.787 1 10264 6811
Intercept[15] 7.083 0.736 5.714 8.577 1 10651 6968
Intercept[16] 7.722 0.874 6.133 9.590 1 10737 6622
Intercept[17] 8.867 1.225 6.832 11.660 1 12393 6346
Difference in Hours Played 0.030 0.022 -0.012 0.073 1 8528 6298
Loneliness During Lockdown 0.398 0.076 0.250 0.551 1 9196 6575
Hours by Loneliness -0.003 0.004 -0.010 0.004 1 8450 6130
Stress
Intercept[1] 0.326 0.449 -0.562 1.196 1 10986 6654
Intercept[2] 0.841 0.435 -0.019 1.694 1 11227 6696
Intercept[3] 1.195 0.430 0.361 2.032 1 11012 6872
Intercept[4] 1.618 0.432 0.773 2.467 1 10573 6620
Intercept[5] 2.010 0.438 1.162 2.874 1 10161 6885
Intercept[6] 2.404 0.446 1.537 3.288 1 9828 6715
Intercept[7] 2.761 0.454 1.878 3.662 1 9734 6706
Intercept[8] 3.183 0.464 2.292 4.102 1 9830 6534
Intercept[9] 3.436 0.472 2.520 4.358 1 9739 6527
Intercept[10] 3.853 0.484 2.902 4.807 1 9712 6620
Intercept[11] 4.203 0.494 3.242 5.193 1 9844 6827
Intercept[12] 4.428 0.500 3.451 5.418 1 9879 6553
Intercept[13] 4.938 0.519 3.928 5.968 1 10134 6701
Intercept[14] 5.272 0.528 4.254 6.313 1 10246 6567
Intercept[15] 5.589 0.537 4.560 6.657 1 10376 6467
Intercept[16] 5.773 0.545 4.718 6.854 1 10411 6242
Intercept[17] 5.976 0.552 4.919 7.061 1 10385 6607
Intercept[18] 6.560 0.583 5.433 7.726 1 10993 6510
Intercept[19] 7.312 0.647 6.088 8.602 1 11868 6512
Intercept[20] 8.123 0.778 6.688 9.743 1 13028 6317
Intercept[21] 9.574 1.183 7.615 12.282 1 13619 6416
Difference in Hours Played 0.025 0.022 -0.017 0.068 1 9362 6687
Loneliness During Lockdown 0.531 0.076 0.383 0.682 1 9656 6624
Hours by Loneliness -0.002 0.004 -0.010 0.005 1 9492 5892

Study 2

Main Model

term Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Depression
Intercept[1] -8.129 1.068 -10.408 -6.226 1 2577 2973
Intercept[2] -5.793 0.802 -7.474 -4.334 1 2310 3417
Intercept[3] -4.232 0.675 -5.668 -3.023 1 2282 3782
Intercept[4] -3.061 0.584 -4.265 -2.001 1 2301 3463
Intercept[5] -2.132 0.528 -3.242 -1.163 1 2381 3789
Intercept[6] -1.193 0.487 -2.196 -0.297 1 2485 4045
Intercept[7] -0.282 0.463 -1.201 0.625 1 2687 3790
Intercept[8] 0.341 0.456 -0.562 1.227 1 2930 4530
Intercept[9] 0.869 0.454 -0.027 1.765 1 3018 4901
Intercept[10] 1.208 0.459 0.319 2.117 1 2996 4677
Intercept[11] 1.564 0.469 0.670 2.497 1 3031 4484
Intercept[12] 1.714 0.474 0.813 2.669 1 3051 4491
Intercept[13] 2.529 0.512 1.573 3.562 1 3242 4971
Intercept[14] 3.320 0.566 2.271 4.490 1 3343 5008
Intercept[15] 4.019 0.612 2.880 5.288 1 3243 4932
Intercept[16] 4.657 0.663 3.443 6.036 1 3516 5085
Intercept[17] 5.212 0.696 3.927 6.654 1 3699 5000
Intercept[18] 5.719 0.738 4.346 7.260 1 3790 5367
Intercept[19] 6.348 0.792 4.871 7.992 1 3672 5842
Intercept[20] 6.737 0.827 5.194 8.463 1 3882 5903
Intercept[21] 8.326 1.069 6.417 10.610 1 4507 6010
Lockdown Period 0.175 0.149 -0.115 0.464 1 13193 5659
Total Hours 0.803 0.265 0.294 1.334 1 5216 5464
Lockdown Period by Hours 0.004 0.167 -0.324 0.331 1 10430 5660
Anxiety
Intercept[1] -4.919 0.732 -6.441 -3.580 1 1309 2574
Intercept[2] -2.981 0.596 -4.206 -1.887 1 1304 2416
Intercept[3] -1.384 0.514 -2.433 -0.447 1 1335 2467
Intercept[4] -0.075 0.476 -1.044 0.832 1 1355 2712
Intercept[5] 0.731 0.472 -0.193 1.663 1 1489 2949
Intercept[6] 1.752 0.497 0.815 2.737 1 1829 3527
Intercept[7] 2.119 0.508 1.162 3.159 1 1960 3663
Intercept[8] 2.922 0.545 1.904 4.067 1 2374 4806
Intercept[9] 3.470 0.580 2.406 4.666 1 2717 4695
Intercept[10] 3.911 0.613 2.795 5.213 1 2957 5078
Intercept[11] 4.669 0.687 3.399 6.106 1 3316 4680
Intercept[12] 5.757 0.780 4.322 7.396 1 3593 4535
Intercept[13] 6.071 0.805 4.622 7.761 1 3579 4912
Intercept[14] 7.095 0.916 5.420 8.998 1 3952 5212
Intercept[15] 7.691 0.997 5.879 9.770 1 4452 5935
Intercept[16] 9.344 1.382 6.971 12.310 1 5682 5964
Lockdown Period 0.074 0.149 -0.219 0.366 1 9433 6185
Total Hours 0.600 0.258 0.102 1.104 1 2958 4535
Lockdown Period by Hours 0.175 0.165 -0.150 0.501 1 7451 6102
Stress
Intercept[1] -7.554 1.016 -9.681 -5.729 1 2328 3654
Intercept[2] -6.072 0.832 -7.770 -4.539 1 2303 3437
Intercept[3] -4.940 0.727 -6.459 -3.609 1 2197 3696
Intercept[4] -3.712 0.637 -5.071 -2.546 1 2121 3600
Intercept[5] -2.555 0.566 -3.741 -1.509 1 2052 3292
Intercept[6] -1.770 0.522 -2.844 -0.798 1 2080 3638
Intercept[7] -0.599 0.478 -1.556 0.311 1 2063 4014
Intercept[8] -0.251 0.474 -1.205 0.654 1 2112 3662
Intercept[9] 0.485 0.473 -0.442 1.405 1 2189 3650
Intercept[10] 1.164 0.484 0.235 2.129 1 2221 3837
Intercept[11] 2.091 0.513 1.118 3.157 1 2236 3979
Intercept[12] 2.738 0.534 1.727 3.856 1 2344 3843
Intercept[13] 3.071 0.546 2.053 4.210 1 2446 3651
Intercept[14] 3.796 0.578 2.724 4.989 1 2594 4246
Intercept[15] 4.805 0.646 3.626 6.148 1 2835 4838
Intercept[16] 5.271 0.680 4.040 6.697 1 3066 4797
Intercept[17] 6.233 0.777 4.843 7.890 1 3614 5721
Intercept[18] 6.700 0.847 5.189 8.492 1 3854 5171
Intercept[19] 7.310 0.945 5.622 9.388 1 4172 5810
Intercept[20] 9.143 1.449 6.699 12.421 1 4399 5888
Lockdown Period -0.073 0.142 -0.345 0.212 1 10153 5768
Total Hours 0.628 0.261 0.120 1.151 1 4154 5345
Lockdown Period by Hours 0.254 0.166 -0.068 0.581 1 8075 6024
Loneliness
Intercept[1] -9.306 1.506 -12.656 -6.720 1 2008 3113
Intercept[2] -6.705 1.022 -8.932 -4.906 1 2207 3303
Intercept[3] -4.850 0.780 -6.500 -3.489 1 2214 3475
Intercept[4] -2.950 0.599 -4.229 -1.867 1 2338 3776
Intercept[5] -2.190 0.542 -3.332 -1.179 1 2363 3763
Intercept[6] -1.181 0.490 -2.173 -0.250 1 2342 3954
Intercept[7] -0.492 0.474 -1.409 0.434 1 2266 3716
Intercept[8] 0.851 0.492 -0.056 1.843 1 2124 3514
Intercept[9] 1.784 0.530 0.811 2.881 1 2061 3407
Intercept[10] 2.995 0.594 1.920 4.225 1 2096 3408
Intercept[11] 5.055 0.730 3.729 6.596 1 2088 4084
Lockdown Period 0.139 0.151 -0.159 0.437 1 10423 5682
Total Hours 0.212 0.259 -0.298 0.718 1 3971 5380
Lockdown Period by Hours 0.008 0.171 -0.330 0.341 1 8264 6196

Moderation Model

term Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Depression
Intercept[1] -1.244 0.659 -2.596 -0.019 1 10853 5950
Intercept[2] -0.069 0.533 -1.107 0.981 1 13733 6713
Intercept[3] 0.880 0.514 -0.111 1.906 1 12442 7269
Intercept[4] 1.190 0.517 0.189 2.236 1 11996 6713
Intercept[5] 1.673 0.531 0.655 2.746 1 11571 6694
Intercept[6] 2.050 0.546 1.001 3.142 1 11302 6751
Intercept[7] 2.340 0.558 1.282 3.477 1 11073 6564
Intercept[8] 2.745 0.573 1.663 3.898 1 10780 6730
Intercept[9] 3.106 0.589 1.988 4.278 1 10749 6856
Intercept[10] 3.293 0.596 2.150 4.486 1 10698 6430
Intercept[11] 3.487 0.605 2.329 4.699 1 10824 6582
Intercept[12] 3.689 0.614 2.523 4.925 1 10863 6528
Intercept[13] 4.051 0.637 2.844 5.320 1 10790 6298
Intercept[14] 4.373 0.657 3.123 5.676 1 10813 6327
Intercept[15] 4.546 0.671 3.264 5.878 1 10826 6406
Intercept[16] 4.817 0.690 3.521 6.191 1 11065 6552
Intercept[17] 5.226 0.713 3.874 6.659 1 11219 6580
Intercept[18] 5.750 0.755 4.319 7.279 1 11672 6783
Intercept[19] 6.090 0.782 4.607 7.672 1 11577 6981
Intercept[20] 7.270 0.988 5.495 9.400 1 12680 6389
Difference in Hours Played -0.008 0.027 -0.061 0.045 1 10144 6248
Loneliness During Lockdown 0.350 0.070 0.219 0.488 1 12083 6958
Hours by Loneliness 0.003 0.003 -0.004 0.009 1 10310 6272
Anxiety
Intercept[1] -0.313 0.494 -1.308 0.645 1 7965 6161
Intercept[2] 0.551 0.479 -0.367 1.477 1 9102 6689
Intercept[3] 1.057 0.486 0.123 2.012 1 8418 6638
Intercept[4] 1.653 0.504 0.683 2.655 1 8014 6817
Intercept[5] 2.003 0.520 1.010 3.033 1 7867 6878
Intercept[6] 2.305 0.532 1.271 3.367 1 7463 6810
Intercept[7] 2.550 0.542 1.495 3.629 1 7495 6791
Intercept[8] 2.885 0.558 1.802 4.007 1 7534 6502
Intercept[9] 3.108 0.571 2.007 4.271 1 7363 6496
Intercept[10] 3.440 0.590 2.315 4.624 1 7586 6775
Intercept[11] 3.726 0.609 2.558 4.963 1 7788 6628
Intercept[12] 4.061 0.634 2.870 5.353 1 8519 6741
Intercept[13] 4.326 0.659 3.076 5.632 1 8676 7136
Intercept[14] 5.320 0.793 3.857 6.994 1 8939 6931
Intercept[15] 6.783 1.187 4.762 9.435 1 10142 6359
Difference in Hours Played 0.017 0.026 -0.033 0.070 1 6094 4348
Loneliness During Lockdown 0.220 0.064 0.095 0.349 1 7270 5731
Hours by Loneliness -0.001 0.003 -0.007 0.005 1 6385 4947
Stress
Intercept[1] -1.867 0.635 -3.168 -0.670 1 9482 6356
Intercept[2] -1.211 0.570 -2.341 -0.093 1 10822 6706
Intercept[3] -0.680 0.541 -1.735 0.385 1 10998 6310
Intercept[4] -0.012 0.520 -1.017 1.017 1 11805 6467
Intercept[5] 0.535 0.514 -0.456 1.560 1 11576 6547
Intercept[6] 0.815 0.514 -0.160 1.838 1 11740 6759
Intercept[7] 1.227 0.514 0.253 2.266 1 11677 6877
Intercept[8] 1.328 0.516 0.342 2.355 1 11563 6483
Intercept[9] 1.688 0.528 0.700 2.754 1 11382 6595
Intercept[10] 2.012 0.544 0.992 3.094 1 11245 6506
Intercept[11] 2.127 0.548 1.093 3.213 1 11215 6400
Intercept[12] 2.478 0.565 1.412 3.601 1 10880 6512
Intercept[13] 2.731 0.575 1.652 3.868 1 11107 6479
Intercept[14] 3.249 0.599 2.120 4.449 1 11070 6837
Intercept[15] 3.835 0.632 2.656 5.102 1 11353 6783
Intercept[16] 4.232 0.671 2.985 5.597 1 11322 6331
Intercept[17] 4.764 0.736 3.398 6.284 1 11337 6748
Intercept[18] 5.262 0.813 3.777 6.937 1 11324 6709
Intercept[19] 6.735 1.201 4.694 9.473 1 11764 6318
Difference in Hours Played 0.050 0.026 0.000 0.101 1 9122 6275
Loneliness During Lockdown 0.207 0.067 0.079 0.337 1 11729 6400
Hours by Loneliness -0.004 0.003 -0.011 0.002 1 9086 6622

Study 3

Main Model

term Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Depression
Intercept[1] -6.862 1.013 -9.014 -5.016 1 2019 3197
Intercept[2] -4.816 0.774 -6.436 -3.431 1 1948 3181
Intercept[3] -3.863 0.677 -5.283 -2.633 1 2001 3409
Intercept[4] -3.062 0.608 -4.315 -1.964 1 1973 3372
Intercept[5] -2.214 0.550 -3.354 -1.230 1 1946 3401
Intercept[6] -1.404 0.508 -2.444 -0.470 1 1919 3687
Intercept[7] -1.096 0.500 -2.122 -0.159 1 1961 3589
Intercept[8] -0.133 0.483 -1.095 0.797 1 2040 3665
Intercept[9] 0.217 0.483 -0.739 1.148 1 2046 3867
Intercept[10] 0.566 0.485 -0.390 1.529 1 2104 3603
Intercept[11] 1.194 0.498 0.226 2.189 1 2129 3369
Intercept[12] 1.815 0.520 0.833 2.885 1 2369 4035
Intercept[13] 2.540 0.559 1.484 3.688 1 2478 3658
Intercept[14] 3.009 0.593 1.920 4.244 1 2510 3996
Intercept[15] 3.534 0.629 2.376 4.837 1 2720 4601
Intercept[16] 4.326 0.695 3.056 5.770 1 2787 4545
Intercept[17] 5.002 0.764 3.617 6.578 1 2982 4866
Intercept[18] 5.651 0.851 4.142 7.438 1 3227 5240
Intercept[19] 6.224 0.932 4.590 8.190 1 3435 5639
Intercept[20] 7.455 1.151 5.444 9.864 1 4266 5797
Lockdown Period 0.091 0.173 -0.244 0.429 1 7024 5940
Total Hours 0.032 0.365 -0.692 0.751 1 2365 3660
Lockdown Period by Hours 0.198 0.189 -0.176 0.573 1 7716 6175
Anxiety
Intercept[1] -5.224 1.048 -7.506 -3.387 1 1139 2856
Intercept[2] -3.532 0.882 -5.441 -1.981 1 1145 2812
Intercept[3] -2.209 0.753 -3.794 -0.849 1 1147 2476
Intercept[4] -1.215 0.696 -2.664 0.073 1 1183 2547
Intercept[5] -0.061 0.657 -1.378 1.184 1 1289 2513
Intercept[6] 0.763 0.649 -0.532 2.025 1 1364 2887
Intercept[7] 1.256 0.653 -0.040 2.540 1 1450 2944
Intercept[8] 2.037 0.668 0.770 3.370 1 1611 3211
Intercept[9] 2.730 0.701 1.367 4.163 1 1791 3259
Intercept[10] 3.418 0.755 2.013 4.970 1 2079 4012
Intercept[11] 3.723 0.778 2.275 5.334 1 2180 3962
Intercept[12] 4.538 0.852 2.962 6.316 1 2424 3695
Intercept[13] 5.272 0.917 3.605 7.243 1 2702 4289
Intercept[14] 6.103 0.996 4.273 8.183 1 2995 4193
Intercept[15] 6.590 1.046 4.703 8.781 1 3217 4602
Intercept[16] 9.072 1.604 6.391 12.609 1 4518 5648
Intercept[17] 11.400 2.333 7.620 16.791 1 4801 4964
Lockdown Period 0.120 0.183 -0.239 0.472 1 9160 6052
Total Hours -0.222 0.386 -0.995 0.521 1 2344 3993
Lockdown Period by Hours -0.079 0.202 -0.476 0.318 1 8277 6488
Stress
Intercept[1] -7.315 1.148 -9.772 -5.280 1 2703 3662
Intercept[2] -6.145 0.980 -8.230 -4.391 1 2683 4218
Intercept[3] -5.381 0.899 -7.287 -3.779 1 2705 4033
Intercept[4] -4.321 0.796 -5.994 -2.897 1 2666 4214
Intercept[5] -3.909 0.757 -5.501 -2.544 1 2591 3903
Intercept[6] -2.572 0.644 -3.908 -1.401 1 2535 3861
Intercept[7] -1.381 0.576 -2.546 -0.304 1 2502 4190
Intercept[8] -0.805 0.559 -1.951 0.263 1 2549 4366
Intercept[9] -0.057 0.552 -1.151 1.032 1 2653 4503
Intercept[10] 0.323 0.550 -0.769 1.418 1 2613 4235
Intercept[11] 0.905 0.559 -0.194 2.040 1 2709 4461
Intercept[12] 1.873 0.590 0.762 3.062 1 2916 4853
Intercept[13] 2.333 0.608 1.178 3.543 1 2998 4974
Intercept[14] 3.048 0.650 1.868 4.416 1 3096 5234
Intercept[15] 3.543 0.692 2.267 4.993 1 3271 5543
Intercept[16] 5.214 0.890 3.589 7.094 1 3713 5037
Intercept[17] 6.157 1.015 4.324 8.254 1 3878 5100
Intercept[18] 7.147 1.185 5.064 9.684 1 3889 4905
Intercept[19] 8.425 1.432 5.911 11.493 1 3844 4634
Intercept[20] 10.027 1.696 7.013 13.708 1 4369 5460
Lockdown Period -0.002 0.175 -0.348 0.346 1 11602 5922
Total Hours -0.128 0.353 -0.822 0.574 1 3946 5263
Lockdown Period by Hours 0.237 0.189 -0.138 0.613 1 9330 6316
Loneliness
Intercept[1] -8.966 1.753 -12.941 -6.081 1 1977 3121
Intercept[2] -7.584 1.390 -10.733 -5.204 1 1995 3694
Intercept[3] -5.730 1.008 -7.900 -3.966 1 1786 3608
Intercept[4] -4.065 0.768 -5.691 -2.714 1 1769 3810
Intercept[5] -2.752 0.639 -4.090 -1.618 1 1619 3201
Intercept[6] -1.673 0.568 -2.865 -0.620 1 1624 3194
Intercept[7] 0.037 0.509 -0.950 1.045 1 1903 3597
Intercept[8] 0.881 0.528 -0.102 1.979 1 1918 3521
Intercept[9] 1.629 0.564 0.590 2.797 1 2060 3891
Intercept[10] 2.927 0.652 1.738 4.281 1 2186 3992
Intercept[11] 5.245 0.858 3.702 7.082 1 2598 4338
Lockdown Period 0.153 0.181 -0.200 0.510 1 7045 5374
Total Hours 0.253 0.342 -0.397 0.949 1 2218 3794
Lockdown Period by Hours -0.090 0.184 -0.458 0.268 1 6485 5733

Moderation Model

term Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Depression
Intercept[1] -1.364 0.973 -3.397 0.444 1 6461 5301
Intercept[2] 0.232 0.757 -1.257 1.729 1 9280 6869
Intercept[3] 0.637 0.738 -0.799 2.095 1 9055 6566
Intercept[4] 1.094 0.726 -0.314 2.545 1 8996 6674
Intercept[5] 1.749 0.736 0.360 3.242 1 8759 6940
Intercept[6] 2.001 0.750 0.585 3.550 1 8695 6876
Intercept[7] 2.262 0.765 0.829 3.824 1 8609 7110
Intercept[8] 2.811 0.797 1.296 4.454 1 8394 6527
Intercept[9] 3.023 0.806 1.496 4.660 1 8441 6499
Intercept[10] 3.234 0.814 1.678 4.884 1 8278 6415
Intercept[11] 3.681 0.836 2.085 5.363 1 8170 6550
Intercept[12] 4.004 0.857 2.365 5.730 1 8059 6643
Intercept[13] 4.465 0.886 2.784 6.273 1 7907 6611
Intercept[14] 4.679 0.895 2.976 6.506 1 7854 6385
Intercept[15] 5.036 0.917 3.314 6.882 1 7870 6497
Intercept[16] 5.305 0.933 3.539 7.206 1 7861 6700
Intercept[17] 5.784 0.977 3.946 7.775 1 8092 6331
Intercept[18] 6.180 1.010 4.275 8.255 1 8362 6513
Intercept[19] 6.694 1.069 4.714 8.885 1 8480 6670
Intercept[20] 7.441 1.198 5.267 9.931 1 9116 6774
Difference in Hours Played 0.014 0.065 -0.117 0.139 1 5558 4870
Loneliness During Lockdown 0.406 0.102 0.207 0.612 1 8407 6330
Hours by Loneliness -0.003 0.008 -0.018 0.012 1 5594 4930
Anxiety
Intercept[1] 0.712 0.723 -0.705 2.146 1 10649 6005
Intercept[2] 1.266 0.729 -0.128 2.738 1 11443 6143
Intercept[3] 1.752 0.747 0.340 3.246 1 10733 6512
Intercept[4] 2.138 0.762 0.697 3.649 1 10313 6596
Intercept[5] 2.365 0.776 0.904 3.901 1 9986 6553
Intercept[6] 2.825 0.808 1.319 4.417 1 9425 6504
Intercept[7] 3.055 0.821 1.512 4.674 1 9247 6597
Intercept[8] 3.464 0.841 1.888 5.146 1 9013 6401
Intercept[9] 3.828 0.864 2.216 5.554 1 9010 6355
Intercept[10] 4.026 0.874 2.367 5.749 1 9116 6766
Intercept[11] 4.242 0.882 2.587 6.001 1 9052 6526
Intercept[12] 4.609 0.900 2.916 6.418 1 8953 6605
Intercept[13] 5.082 0.939 3.303 6.945 1 9416 6693
Intercept[14] 5.497 0.975 3.671 7.473 1 9626 6532
Intercept[15] 7.229 1.350 4.836 10.157 1 11167 5835
Difference in Hours Played -0.024 0.073 -0.171 0.115 1 5228 5204
Loneliness During Lockdown 0.312 0.097 0.126 0.503 1 8942 6273
Hours by Loneliness 0.002 0.008 -0.014 0.020 1 5310 5103
Stress
Intercept[1] -1.281 0.807 -2.906 0.271 1 6995 6070
Intercept[2] -0.452 0.743 -1.945 0.983 1 8609 6642
Intercept[3] -0.023 0.723 -1.470 1.402 1 8587 6431
Intercept[4] 0.420 0.715 -0.995 1.820 1 8235 6887
Intercept[5] 0.611 0.712 -0.784 2.019 1 8351 6816
Intercept[6] 1.027 0.707 -0.347 2.423 1 8324 7044
Intercept[7] 1.251 0.710 -0.132 2.627 1 8185 6848
Intercept[8] 1.612 0.718 0.217 3.035 1 8109 6684
Intercept[9] 1.894 0.725 0.498 3.338 1 8095 6717
Intercept[10] 2.181 0.742 0.743 3.666 1 8229 7112
Intercept[11] 2.469 0.755 1.002 3.976 1 8026 6840
Intercept[12] 2.981 0.777 1.466 4.562 1 7995 6887
Intercept[13] 3.136 0.785 1.614 4.735 1 7960 6821
Intercept[14] 3.564 0.802 2.030 5.214 1 7619 6802
Intercept[15] 3.858 0.821 2.288 5.533 1 7432 6546
Intercept[16] 4.650 0.879 2.931 6.434 1 7397 6530
Intercept[17] 5.032 0.920 3.254 6.889 1 7749 6433
Intercept[18] 5.526 0.976 3.642 7.477 1 8033 6598
Intercept[19] 6.269 1.100 4.205 8.498 1 8661 6772
Difference in Hours Played -0.004 0.076 -0.157 0.138 1 5265 5555
Loneliness During Lockdown 0.284 0.093 0.104 0.469 1 7989 6554
Hours by Loneliness 0.000 0.009 -0.017 0.017 1 5281 5921

Prior Sensitivity Check

The below plots show how sensitive the parameter estimates and Bayes factors are to different specifications of priors. The models reported in text had the following priors:

DAS Main Models Loneliness Main Model Moderation Models
Intercept Student-t(3, 0, 2.5) Student-t(3, 0, 1.5) Student-t(3, 0, 3)
Beta Normal(0, 1) Normal(0, 1) Normal(0, 1)
SD Exponential(1) Exponential(1)

For each model we fitted 10 models in total (including the reported model) with different prior specifications for the Beta (slope) terms. For the main and moderation models these ranged from Normal(0, 0.2) to Normal(0, 2) in steps of 0.2, while for the difference models these ranges from Normal(0, 0.1) to Normal(0, 1) in steps of 0.1.

In the plots below each model is reported showing the sensitivity of both the parameter estimates for each term in the model and for the Bayes factor in support of the point null (i.e. a parameter estimate of 0) relative to the alternative model (i.e. a parameter estimate of \(\neq\) 0). In all cases, the standard deviation for the prior on the beta parameter for the models reported in text is indicated with a vertical dotted line.

Generally, these plots show that the models are not overly sensitive to the prior specification, giving consistent results for the parameter estimates and Bayes factors regardless of whether the beta terms have larger or smaller standard deviations than the models reported.

Study 1

Main Models

There is general agreement between models regardless of the magnitude of the standard deviation for the beta prior for the parameter estimates. While the Bayes factor generally increases as the standard deviation for the beta prior increases, this trend is consistent and conclusions do not change across prior specifications.

Moderation Model

As before, conclusions do not change depending upon the magnitude of the standard deviation for the beta prior.

Study 2

Main Model

As before, conclusions do not change depending upon the magnitude of the standard deviation for the beta prior.

Moderation Model

As before, conclusions do not change depending upon the magnitude of the standard deviation for the beta prior.

Study 3

Main Model

As before, conclusions do not change depending upon the magnitude of the standard deviation for the beta prior.

Moderation Model

As before, conclusions do not change depending upon the magnitude of the standard deviation for the beta prior.

Prior Predictive Check

All plots show the density of the observed data (dark lines) plotted against 100 random replications from the priors. In all cases relatively informative priors were used to allow for proper inferences to be draws using Bayes factors. Generally the priors allow for a wide range of effects that differ from the observed data, so are thus not too restrictive. For the main models the replications are close to the observed data. The moderation models show that while the observed data is plausible given the priors, the priors perhaps allow too much weight to large effects and too little weight to middling effects. That said, the reliability of the models is largely dependent on the posterior predictive checks, below.

Study 1

Main Model

Moderation Model

Study 2

Main Model

Moderation Model

Study 3

Main Model

Moderation Model

Anxiety

Posterior Predictive Check

All plots show the density of the observed data (dark lines) plotted against 100 random replications from the posterior. Generally, the posterior approximates the observed data very well. Thus the model estimates are likely to be reliable in all instances.

Study 1

Main Model

Moderation Model

Study 2

Main Model

Moderation Model

Study 3

Main Model

Moderation Model